Theorem eldprdi 19932

Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
eldprdi.0	⊢ 0 = (0g‘𝐺)
eldprdi.w	⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
eldprdi.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
eldprdi.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
eldprdi.3	⊢ (𝜑 → 𝐹 ∈ 𝑊)

Assertion

Ref	Expression
eldprdi	⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))

Distinct variable groups:   ℎ,𝐹   ℎ,𝑖,𝐺   ℎ,𝐼,𝑖   0 ,ℎ   𝑆,ℎ,𝑖

Allowed substitution hints:   𝜑(ℎ,𝑖)   𝐹(𝑖)   𝑊(ℎ,𝑖)   0 (𝑖)

Proof of Theorem eldprdi

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldprdi.1	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		eldprdi.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊)
3		eqid 2731	. . 3 ⊢ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)
4		oveq2 7354	. . . 4 ⊢ (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹))
5	4	rspceeqv 3595	. . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
6	2, 3, 5	sylancl 586	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))
7		eldprdi.2	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
8		eldprdi.0	. . . 4 ⊢ 0 = (0g‘𝐺)
9		eldprdi.w	. . . 4 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }
10	8, 9	eldprd 19918	. . 3 ⊢ (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
11	7, 10	syl 17	. 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
12	1, 6, 11	mpbir2and 713	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395   class class class wbr 5089  dom cdm 5614  ‘cfv 6481  (class class class)co 7346  Xcixp 8821   finSupp cfsupp 9245  0_gc0g 17343   Σ_g cgsu 17344   DProd cdprd 19907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-ixp 8822  df-dprd 19909

This theorem is referenced by:  dprdfsub  19935  dprdf11  19937  dprdsubg  19938  dprdub  19939  dpjidcl  19972